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Theorem cnvcnv 5126
Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
cnvcnv  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )

Proof of Theorem cnvcnv
StepHypRef Expression
1 relcnv 5051 . . . . 5  |-  Rel  `' `' A
2 df-rel 4696 . . . . 5  |-  ( Rel  `' `' A  <->  `' `' A  C_  ( _V 
X.  _V ) )
31, 2mpbi 199 . . . 4  |-  `' `' A  C_  ( _V  X.  _V )
4 relxp 4794 . . . . 5  |-  Rel  ( _V  X.  _V )
5 dfrel2 5124 . . . . 5  |-  ( Rel  ( _V  X.  _V ) 
<->  `' `' ( _V  X.  _V )  =  ( _V  X.  _V ) )
64, 5mpbi 199 . . . 4  |-  `' `' ( _V  X.  _V )  =  ( _V  X.  _V )
73, 6sseqtr4i 3211 . . 3  |-  `' `' A  C_  `' `' ( _V  X.  _V )
8 dfss 3167 . . 3  |-  ( `' `' A  C_  `' `' ( _V  X.  _V )  <->  `' `' A  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
) )
97, 8mpbi 199 . 2  |-  `' `' A  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
10 cnvin 5088 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
11 cnvin 5088 . . . 4  |-  `' ( A  i^i  ( _V 
X.  _V ) )  =  ( `' A  i^i  `' ( _V  X.  _V ) )
1211cnveqi 4856 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  `' ( `' A  i^i  `' ( _V  X.  _V )
)
13 inss2 3390 . . . . 5  |-  ( A  i^i  ( _V  X.  _V ) )  C_  ( _V  X.  _V )
14 df-rel 4696 . . . . 5  |-  ( Rel  ( A  i^i  ( _V  X.  _V ) )  <-> 
( A  i^i  ( _V  X.  _V ) ) 
C_  ( _V  X.  _V ) )
1513, 14mpbir 200 . . . 4  |-  Rel  ( A  i^i  ( _V  X.  _V ) )
16 dfrel2 5124 . . . 4  |-  ( Rel  ( A  i^i  ( _V  X.  _V ) )  <->  `' `' ( A  i^i  ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) ) )
1715, 16mpbi 199 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  ( A  i^i  ( _V  X.  _V )
)
1812, 17eqtr3i 2305 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) )
199, 10, 183eqtr2i 2309 1  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    i^i cin 3151    C_ wss 3152    X. cxp 4687   `'ccnv 4688   Rel wrel 4694
This theorem is referenced by:  cnvcnv2  5127  cnvcnvss  5128  structcnvcnv  13159  strfv2d  13178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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